If ` sigma _x =10 ,` ` sigma _y =10 ,` then the value of `Cov(x,y)` is

Options

A. `100 `
B. ` sqrt(10)`
C. ` 10 `
D. ` 0`

Answer

To find the value of Cov(x, y) given that σ_x = 10 and σ_y = 10, we can follow these steps:

### Step 1: Understand the relationship between covariance and correlation
The formula for the correlation coefficient (ρ) between two variables x and y is given by:
\[ \rho_{x,y} = \frac{Cov(x,y)}{\sigma_x \sigma_y} \]
Where:
- Cov(x, y) is the covariance between x and y.
- σ_x is the standard deviation of x.
- σ_y is the standard deviation of y.

### Step 2: Identify the given values
From the problem, we know:
- σ_x = 10
- σ_y = 10

### Step 3: Analyze the correlation
Since both σ_x and σ_y are positive, we can conclude that x and y are positively correlated. The maximum value of the correlation coefficient (ρ) is 1 when x and y are perfectly positively correlated.

### Step 4: Substitute the values into the correlation formula
Assuming the maximum correlation (ρ = 1), we can substitute the values into the correlation formula:
\[ 1 = \frac{Cov(x,y)}{10 \times 10} \]

### Step 5: Solve for Cov(x, y)
Now we can rearrange the equation to solve for Cov(x, y):
\[ Cov(x,y) = 1 \times (10 \times 10) \]
\[ Cov(x,y) = 100 \]

### Final Answer
The value of Cov(x, y) is 100.

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